After noon, the number of people in a major department store grows steadily until 9 p.m. If the equation y = 118 + 23x represents the number of people in the store x hours after no... After noon, the number of people in a major department store grows steadily until 9 p.m. If the equation y = 118 + 23x represents the number of people in the store x hours after noon. a. How many people were in the store at noon? b. At what rate is the number of shoppers growing? c. When were there 279 shoppers in the store?
Understand the Problem
The question is about a growth model for the number of people in a store after noon. It asks for initial conditions (number of people at noon), the growth rate of shoppers, and a specific time when a certain number of shoppers are present in the store. This requires applying the given equation to extract the necessary information.
Answer
a. 118 people b. 23 people per hour c. 7 p.m.
Answer for screen readers
a. 118 people
b. 23 people per hour
c. 7 p.m.
Steps to Solve
- Identify the initial number of people at noon
To find the number of people in the store at noon ($x = 0$), substitute $x$ with 0 in the equation:
[ y = 118 + 23(0) = 118 ]
So, there were 118 people in the store at noon.
- Determine the growth rate of shoppers
The growth rate is represented by the coefficient of $x$ in the equation. Here, 23 is the coefficient, which means:
The number of shoppers is growing at a rate of 23 people per hour.
- Find when there were 279 shoppers in the store
To find out when there were 279 shoppers, set $y$ equal to 279 and solve for $x$:
[ 279 = 118 + 23x ]
Subtract 118 from both sides:
[ 279 - 118 = 23x \implies 161 = 23x ]
Now divide both sides by 23:
[ x = \frac{161}{23} \approx 7 ]
Since $x$ represents hours after noon, add 7 hours to noon:
[ \text{Noon} + 7 \text{ hours} = 7 \text{ p.m.} ]
a. 118 people
b. 23 people per hour
c. 7 p.m.
More Information
The equation represents a linear growth model, where the constant 118 indicates the initial number of people in the store, and the 23 indicates the number of additional people entering every hour. The linear model helps in predicting population growth over a specific period.
Tips
- Confusing the growth rate with the initial number. Remember, the initial number is the value of $y$ when $x=0$.
- Miscalculating the time when a specific number of shoppers is reached. Ensure to correctly isolate $x$ and remember to convert hours after noon to the correct time.
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